PHYS-2212 LAB Coulomb s Law and the Force between Charged Plates

Transcription

1 PHYS-2212 LAB Coulomb s Law and the Force between Charged Plates Objectives To investigate the electrostatic force between charged metal plates and determine the electric permittivity of free space, ε o, (assumed to be the medium between the plates). Introduction In this experiment, a sensitive electrostatic balance is used to measure the force of attraction F between two parallel but oppositely-charged metal plates of equal area A. By connecting these plates to the positive and negative terminals of a power supply, the plates become uniformly charged (one positive and the other negative) electrodes. These oppositely-charged parallel electrodes separated by a small distance form a parallel-plate capacitor. Being conductors, the charges on both plates appear on the inner surfaces due to attractive force between these opposite charges on the plates. When the parallel capacitor plates are modeled as oppositely-charged parallel planes, with the total charge (q) on the inside surface of the positive (lower) plate, the electric field at points very close to the surface of this plate is approximately: q E+ =. Subsequently, the magnitude of the electrostatic force exerted on the negativelycharged upper plate by the lower plate is given 2Aε o by: F = qe+ Eqn. 1 2 q = 2Aε In this experiment, the charge q is directly related to the voltage or potential difference (ΔV) between the capacitor plates. According to the theory, within the voltage interval (ΔV) between two oppositely-charged parallel electrodes separated by displacement d, there exists a uniform electric field, E. with magnitude given as: E = ΔV d Eqn. 2 In the space between the capacitor plates, the resultant electric field is the superposition of the electric fields due to the positively- and negatively-charged plates and points from the positive plate to the negative plate. For a plate separation that is very small compared to the surface dimensions of the plates, the magnitude of the resultant electric field is: o 2007 mokafor GEORGIA PERIMETER COLLEGE Page 1 of 8

2 E = q Aε Eqn. 3 Combining Eqn. 2 and Eqn. 3, the relationship between the charge and potential difference becomes: Substituting for q into Eqn. 1 yields: o A q = ε o ΔV d Eqn. 4 F A = ε 2d o 2 ( ΔV ) 2 Eqn. 5 This experiment is designed around the Sargent-Welch current balance equipment with the coulomb balance attachment. The attachment consists of two square metal plates with wire supports for mounting the plates. The plates are separated with spacer blocks of known even thickness. The spot of a laser beam reflection from the balance mirror to a distant screen is marked to correspond to the desired spacing between the plates. Thus, the plates can always be returned to the same preset spacing by adjusting the laser beam to the marked spot on the screen. The distance from the mirror to the screen should be fairly large for higher sensitivity. The reflected laser beam spot on the screen is displaced through an angle twice the angle of rotation of the mirror. Therefore, a long mirror-to-screen distance significantly increases the effects of small rotations of the mirror. The swinging motion of the beam assembly is damped by a conducting blade that oscillates within the magnetic field of a permanent magnet. Apparatus Electrostatic balance, calibrated weights, (spirit) level, ( V) dc power supply, ( V dc) digital voltmeter (DMM), 10,000-Ω resistor 10 W, ⅛-inch spacer block, He-Ne laser, Vernier caliper, micrometer caliper, masking tape mokafor GEORGIA PERIMETER COLLEGE Page 2 of 8

3 Laser DMM Power supply Calipers Resistor Electrostatic balance Spacer block Figure 1: Electrostatic Balance setup Experimental Procedure 1. Review the overview of the electrostatic balance presented in the introduction. Identify the components of the experimental setup shown in Figure 1 above and the essential parts of the electrostatic balance labeled in Figure Use a micrometer caliper to determine the average thickness of the ⅛-inch spacer block. 3. Use Vernier calipers to measure and record the length and width of the two metal plates (A) used in this apparatus. For measuring the dimensions of the plates, you may have to remove the movable plate out of the beam balance and the fixed plate from the base support posts. 4. Carefully lift and remove the balance beam assembly (B). Place a level (provided) on the base of the electrostatic balance and use the two thumbscrews (S) near the front edge to level the balance. 5. Gently replace the beam assembly onto the knife-edge bearing surfaces, using the tapered screw points on the rotating lifting shaft (J) mokafor GEORGIA PERIMETER COLLEGE Page 3 of 8

4 6. Make sure that the movable metal plate that is attached to the beam balance assembly and the fixed plate that is mounted on the base support posts (C) are exactly parallel. Note that there are four sets of adjusting screws for positioning the plates. B M D H G A J S C Figure 2: Electrostatic balance to measure Coulomb s force between charged plates 7. Use the beam lift (J) to adjust the beam assembly and hence correctly position the movable plate so that this upper plate covers the lower fixed plate, with the edges matching all the way round. Tighten the screws after adjusting the plates. 8. After positioning the beam assembly correctly, check and adjust the vertical and horizontal counterweights. The vertical counterweights (G) usually need no further adjustments and are used to set the center of gravity of the balance beam below the knife edge pivots. The resulting fairly low center of gravity provides the mechanical restoring force necessary to overcome the destabilizing effect caused by the attractive electrostatic force between the plates at the limiting conditions of the experiment. The horizontal counterweights (D) are adjusted until the beam assembly is balanced and pivots freely on its knife edges. While pivoting freely on the knife edges, the damping blade (H) should also swing freely inside the groove, without touching any damping magnets. Otherwise gently press the blade sideways, as needed, to ensure free motion of the damping blade. 9. Connect the positive terminal of the power supply to one end of the 10,000-Ω protective resistor provided. Connected the other end of the resistor to one of the connection terminals on the support post holding the fixed plate of the electrostatic balance. Connect 2007 mokafor GEORGIA PERIMETER COLLEGE Page 4 of 8

5 the negative terminal of the power supply to the movable plate through one of the terminals at each knife edge support. Finally, connect the voltmeter (DMM) directly across the terminals of the power supply, positive to positive, negative to negative. When the plates are not touching each other, the voltmeter measures the voltage across the plates. Figure 3: The 10,000-Ω protective resistor 10. STOP!!! Ask your lab instructor to inspect your setup. After your instructor has inspected the wiring and approved your setup, you may continue with the procedure. 11. With the ⅛-inch spacer inserted at the center of both plates, place another circular mass on top of the movable plate (at its center) to hold the plates together. Check the beam assembly and ensure that the plates are exactly parallel and evenly separated by the thickness of the spacer. Figure 4: Pair of electrostatic plates 12. Mount the laser in front of the electrostatic balance so that the laser beam is reflected from the mirror (M) on the balance to a screen approximately 2 m away from the mirror. With the spacer block held in the middle, between the plates, mark the position of the laser light beam on the screen as the reference mark for the experimental measurements. At this reference mark, the spacing between the plates should be exactly equal to the thickness of the spacer. 13. Remove the circular mass that holds the top (movable) plate down and also remove the spacer from between the plates. Place a mass of 70 mg on top of the movable plate, at its 2007 mokafor GEORGIA PERIMETER COLLEGE Page 5 of 8

6 center. Adjust the horizontal counterweights to tilt the top plate upward until the laser beam returns to the reference mark on the screen. 14. Remove the 70-mg mass from the top of the movable plate. This top plate should then tilt upward such that the laser beam shifts above the reference mark on the screen. With the voltage control knob set at zero, switch on the power supply and gradually increase the voltage until the beam assembly is balanced with the laser beam at the reference mark on the screen. Record the voltage reading at which the laser beam reaches the reference mark exactly. Obtain two more independent measurements for this 70-mg mass, starting from zero voltage each time, and find the average voltage. 15. Repeat procedure steps 12 and 13 for each of the following masses: 50 mg, 30 mg, and 20 mg. Record the voltages obtained from three independent measurements with each mass and find the average voltage values. Analysis of Data 16. Determine the average surface area and thickness of each plate from three independent measurements of its length, width, and thickness as recorded in your lab notebook. Average length of plate = Area of plate (A) = m 2 Average width of plate = Average thickness of plate = 17. Determine the average thickness of the spacer block from the three independent measurements using the micrometer calipers shown in Table 1. Table 1: Thickness of spacer block Thickness of the spacer block Micrometer readings Trial 1 Trial 2 Trial 3 Average values Zero reading Reading with spacer Thickness (d) of the ⅛-inch spacer block 2007 mokafor GEORGIA PERIMETER COLLEGE Page 6 of 8

7 18. From the voltage measurements obtained for the each mass removed from the top movable plate and plate separation, shown in Table 2, calculate the electrostatic force on the movable plate and the square of the average voltage for each data set and enter these results in Table 3. Table 2: Voltage measurements for masses placed on the top plate Mass placed on the top plate Voltage Measurements Trial 1 Trial 2 Trial 3 Average Voltage 19. Using data Table 3, plot a linear graph of the forces (F) as ordinates against the matching values of the square voltages (ΔV) 2 as abscissa. Table 3: Graph data for Force on plate vs. square voltages Force on the top plate (N) Square of average voltages (V) mokafor GEORGIA PERIMETER COLLEGE Page 7 of 8

8 20. Since there is zero force at zero voltage, the best line of fit through the plotted points must pass through the origin (0, 0). Draw a regression line through your points. Calculate the slope of this graph of F vs. (ΔV) 2. Slope (with the ⅛-inch separation) = N/V According to theory, the force F with which one plate attracts the other is related to the potential difference (ΔV) between the plates of surface area A separated by a displacement d as in Eqn. 5: A o 2 F = ε ( ΔV ) 2 2d Based on this equation, use the slope and calculate the measured value of ε o, electric permittivity of free space (medium between the plates is actually air). Value of ε o = F/m 22. Compare this measured value of ε o with the accepted value of permittivity of free space by finding the percent error. Discuss how well the data plots of the graph match up with the predictions of the theory, despite the poor level of precision of this experiment. 23. According to theory, comment on the significance of the graph and its calculated slope when compared to the theoretical model. 24. Considering the two conducting plates as a parallel-plate capacitor, use the acquired data for the plate area, the spacing between the plates, and the calculated value of the permittivity to compute the capacitance of your parallel-plate capacitor. 25. Suppose that the square lower plate were placed on a flat surface without the supporting wires. Use the average thickness of the aluminum plates and the density of aluminum as 2.7 g/cm 3 to estimate the mass of the plate. Hence, use Eqn. 5 to find the minimum potential difference between the plates that will be sufficient to overcome the force of gravity and lift this lower plate off the table when the spacing between the plates is equal to the measured thickness of your ⅛-inch spacer block. Comment on this calculated value of the minimum potential difference necessary to just lift the lower plate. Is this result reasonable and feasible? 2007 mokafor GEORGIA PERIMETER COLLEGE Page 8 of 8

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